MAT 156 Lab 10 Topic 1 : Integration by Parts with(student): The steps to perform integration by parts on int(f(x),x) are : 1) choose a function u 2) factor f as f = u dv 3) Integrate dv to find v and differentiate u to find du 4) Replace 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 with 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 Maple has a command that does steps 2, 3 and 4. You must still choose the function u. The command is intparts(f, u) Example 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 We can try u = 2x+3 Fint:=Int(exp(2*x)*(2*x+3), x);

Gint:=intparts(Fint, 2*x+3);

This is now an easier integral so the integration by parts was successful. value(Gint);

Assignment Part 1 : Perform the integration by parts in the integral LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KC1JI21vR0YkNi1RJiZpbnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktRiM2Ji1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjZRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEjMTBGJ0Y1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GMjYuUTEmSW52aXNpYmxlVGltZXM7RicvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RidGNUY4RjtGPUY/RkFGQ0ZFRkdGSi1GIzYlLUYsNiVRI2xuRicvRlVGOkY1LUYyNi1RMCZBcHBseUZ1bmN0aW9uO0YnRjVGOEY7Rj1GP0ZBRkNGRUZHRkotSShtZmVuY2VkR0YkNiQtRiM2I0ZRRjVGK0YrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRmNwLyUqbGluZWJyZWFrR1ElYXV0b0YnLUYyNi1RMCZEaWZmZXJlbnRpYWxEO0YnRjVGOEY7Rj1GP0ZBRkNGRUZHRkpGUUYr. Verify your answer by asking Maple to integrate directly. Many times we are forced to integrate by parts repeatedly. We do this one step at a time. Example: Calculate the 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. Lint:=Int(exp(2*x)*cos(3*x), x);

Now do integration by parts with u = exp(2x) Mint:=intparts(Lint, exp(2*x));

We define the new integral to be Nint: Nint:=Int(2/3*exp(2*x)*sin(3*x), x);

Do integration by parts again with u = exp(2x) Oint:=intparts(Nint, exp(2*x));

Here we see that we got back the integral we started with with coefficient 4/9 in front. The result is Lint:=1/3*exp(2*x)*sin(3*x)-Oint;

Lint:=(1+4/9)^{-1}*(1/3*exp(2*x)*sin(3*x)+2/9*exp(2*x)*cos(3*x)); We can check the answer by redefining the integral and integrating directly: lint:=Int(exp(2*x)*cos(3*x), x);

value(lint); Assignment Part 2 : Perform 2 integrations by parts to compute the integral 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. Check your answer with direct integration. Assignment Part 3 : Evaluate the integral 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. If you try to compute the integral directly you will see that Maple gets stuck and does not know what to do. You will need two integrations by parts and the second is tricky. Ask Maple to simplify the second integrand and to use LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYmUSJ1RicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUTJDaGFyYWN0ZXJTdHlsZTI2MkYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIj1GJ0Y3L0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiZRInhGJ0Y0RjdGOkYr. This shows that sometimes we need to help Maple in its work.